Ad (1):
Proposition REF , together with the observation that \(Q^{\prime }\)  is a left adjoint, readily implies that \(Q^{\prime }\)  is a left Quillen functor.
The fact that this is a Quillen equivalence follows from the existence of the natural weak equivalence \(q^{\prime } \colon Q^{\prime } \overset{\sim }{\longrightarrow }1_{{{K}}_\infty }\) .
Formally, this implies that the composition of \(Q^{\prime }\)  by the left Quillen equivalence \(1_{{{K}}_\infty } \colon {{K}}_\infty ^p \rightarrow {{K}}_\infty ^i\)  is weakly equivalent to the identity functor on \({{K}}_\infty \) .
Thus, the statement follows from the two-out-of-three property of Quillen equivalences [1]} and Corollary REF .
